Pointing Chandra Toward the Extreme Ultraviolet Fluxes of Very Low-Mass Stars

TL;DR

This paper presents a novel method using Chandra X-ray data to estimate the EUV emission of very low-mass stars, crucial for understanding planetary atmosphere evolution, especially for stars too faint for direct EUV observation.

Contribution

The study introduces a new approach to infer EUV fluxes of ultra-cool dwarf stars from X-ray observations, enabling better modeling of their atmospheric impact.

Findings

The method can estimate EUV spectra within a factor of 2-4 for wavelengths below 400-500 Å.

A shallow emission measure distribution slope ($DEM \,\propto T^{3/2}$) fits the data well.

Model uncertainties increase significantly at longer wavelengths and lower temperatures.

Abstract

The X-ray and EUV emission of stars plays a key role in the loss and evolution of the atmospheres of their planets. The coronae of dwarf stars later than M6 appear to behave differently to those of earlier spectral types and are more X-ray dim and radio bright. Too faint to have been observed by the Extreme Ultraviolet Explorer, their EUV behavior is currently highly uncertain. We have devised a method to use the Chandra X-ray Observatory High Resolution Camera to provide a measure of EUV emission in the 50-170 \AA\ range and have applied it to the M6.5 dwarf LHS 248 in a pilot 10 ks exposure. Analysis with model spectra using simple, idealised coronal emission measure distributions inspired by an analysis of Chandra HETG spectra of the M5.5 dwarf Proxima Cen and results from the literature, finds greatest consistency with a very shallow emission measure distribution slope, $DEM \propto T^{3/2}$ or shallower, in the range $\log T=5.5$-$6.5$. Within $2\sigma$ confidence, a much wider range of slopes can be accommodated. Model spectra constrained by this method can provide accurate (within a factor of 2-4) synthesis and extrapolation of EUV spectra for wavelengths $<400$-500 \AA. At longer wavelengths models are uncertain by an order of magnitude or more, and depend on the details of the emission measure distribution at temperatures $\log T < 5.5$. The method is sensitive to possible incompleteness of plasma radiative loss models in the 30 170 \AA\ range for which re-examination would be warranted.

Figures

Tables

Table 1. Table 1: Details of Chandra observations used in this study

Target	ObsID	Instrument	$t_{\mathrm{start}}$	$t_{\mathrm{stop}}$	Exp. (s)
LHS 248	20165	HRC-S	2018-05-22 07:17:04	2018-05-22 10:30:12	10065
Proxima	12360	HETG+ACIS-S	2010-12-13 00:19:15	2010-12-13 22:56:56	78234
Proxima	19708	LETG+HRC-S	2017-05-31 16:25:42	2017-06-01 05:28:58	44349
Proxima	20073	LETG+HRC-S	2017-05-15 23:29:05	2017-05-16 11:22:33	40245
Proxima	20080	LETG+HRC-S	2017-05-18 09:21:04	2017-05-19 00:28:41	51851
Proxima	20084	LETG+HRC-S	2017-06-03 04:45:24	2017-06-03 13:51:01	29479

Equations

$$DEM(T)=n_e^2(T)\frac{dV(T)}{d\log T}.$$

$$\Phi (T)$$

$$=\Phi (T_{min}){\left(\frac{T}{T_{min}}\right)}^{{\alpha }_{LC}};$$

$$=\Phi (T_{peak}){\left(\frac{T}{T_{peak}}\right)}^{{\alpha }_{UC}};$$

$$=\Phi (T_{sh}){\left(\frac{T}{T_{sh}}\right)}^{{\alpha }_{HT}};$$

$$\log T_{min}$$

$$\log T_{sh}-\log T_{peak}$$

$${\alpha }_T$$

$${\alpha }_{UC}$$

$${\alpha }_{HT}$$

$$6.5\le \log T_{peak}\le 6.9$$

$$0.4\le \log \Phi (T_{peak})-\log \Phi (T_{min})\le 3.$$

Full text

Stars: activity, flare, late-type, coronae, low-mass — Sun: X-rays — X-rays: stars — planets and satellites: atmospheres

††software: PINTofALE (Kashyap.Drake:00), CIAO (Fruscione et al., 2006)

1 Introduction

Magnetic activity on stars like the Sun with outer convection zones is manifest through energetic photon and particle radiation in the form of UV to X-ray emission from hot coronal plasma and a hot magnetized wind. Both photon and particle emissions have a stochastic component where stored magnetic potential energy is more impulsively released in flares and coronal mass ejections. Magnetic activity is driven by stellar rotation and so gradually declines with time as stars lose angular momentum to stellar winds (e.g. Skumanich, 1972; Wright et al., 2011; Garraffo et al., 2018). Understanding of the physics of stellar coronae remains incomplete, and its study continues to be a cottage industry in modern astrophysics.

The growing realization that planetary systems are extremely common around Sun-like stars has led to a resurgence in interest in stellar magnetic activity. Exoplanets have catalyzed a shift in focus from coronal physics itself to the effects of energetic stellar radiation on planetary atmospheres and environments. Both UV–X-ray emission and stellar winds are agents of atmospheric destruction. In our own solar system, the action of the solar wind stripped copious amounts of water from the atmosphere of Mars during the Noachian period about 4 Gyr ago (e.g. Jakosky et al., 2015). The roles of X-ray and EUV radiation in exoplanet atmospheric evaporation has been highlighted in several recent studies (e.g. Penz & Micela, 2008; Murray-Clay et al., 2009; Sanz-Forcada et al., 2010; Owen & Jackson, 2012; Owen & Wu, 2013; Chadney et al., 2015; Owen & Wu, 2016), and there is now convincing evidence that planetary envelopes have been reduced and evaporated by the host star EUV and X-ray emission (e.g. Sanz-Forcada et al., 2011; Owen & Wu, 2013). Planets which have received higher integrated doses of EUV and X-ray exposure tend to be smaller objects that appear to have lost part or all of their envelopes.

Based on a simplifying assumption that EUV and X-ray emission decline with time (due to stellar spin down) at the same rate, Owen & Jackson (2012) find that X-rays dominate mass loss during early times when stellar activity is very high. At later times, EUV emission dominates. In their modelling, Owen & Jackson (2012) adopted the assumption of lockstep decline in EUV and X-ray fluxes because, in their words, “The evolution of the EUV luminosity is still observationally rather unclear”. Current evidence seems to point to X-ray decline being more rapid than EUV decline (e.g. Sanz-Forcada et al., 2011; Chadney et al., 2015), but it is now apparent that a deeper understanding of how stellar EUV and X-ray fluxes relate to each other and how they change with time is key to understanding planetary atmospheric evolution.

The “Unobservable Extreme Ultraviolet” is a phrase coined some decades ago representing the largely mistaken belief that the ISM would be so opaque at EUV wavelengths that nothing could be observed. It is apt today because there is currently no facility capable of observing cosmic (i.e. non-solar) EUV emission. While the longer wavelengths of the EUV bandpass (here loosely defined as 100–912 Å) are indeed essentially inaccessible for all but the most nearby stars, the Extreme Ultraviolet Explorer (EUVE) did accumulate spectra of a number of either nearby or particularly active late-type stars in the 70-750 Å range (e.g. Drake, 1996; Bowyer et al., 2000). These data have been used by different authors to examine the EUV flux and how it can be estimated using different proxies, such as X-ray flux (Sanz-Forcada et al., 2011) or Ly$\alpha$ emission (Linsky et al., 2014; Youngblood.etal:16). However, these studies have been limited to stars of spectral type M5 and earlier, for the simple fact that later types were too faint for EUVE to observe.

The remarkable discoveries of Proxima b (Anglada-Escudé et al., 2016) and four additional planets around TRAPPIST-1, making a total of seven currently known (Gillon et al., 2017), has highlighted the case for M dwarfs being favorable candidate host stars with nearby Earth-like exoplanets amenable to detailed study. While M dwarfs are the most numerous stars in the Galaxy, and will likely host the largest number of conveniently nearby exoplanets, their magnetic activity represents a proportionally larger fraction of their bolometric output than earlier type stars, and longer spin down timescales means this output stays higher for longer. Habitable zone planets around M dwarfs are then even more susceptible to EUV, X-ray, and stellar wind destruction.

The EUV flux from late-type stars originates from the chromosphere, the transition region, and the corona (e.g. Drake, 1998; Linsky et al., 2014), and it is necessary to have an accurate description of all of these regions to understand the EUV radiative output of stars. Late-type M dwarfs (spectral types $>$M5) such as TRAPPIST-1 (M8 V) present a difficult case for the use of proxies to infer EUV radiation output. The effective temperature of an M8 dwarf is only about 2500 K and its atmosphere is highly neutral. These stars are fully convective and it has been conjectured that their dynamo activity is fundamentally different. Wright & Drake (2016) have shown that fully convective stars down to type M5.5V do follow the same rotation-activity relation as earlier types. However, relative to bolometric luminosity both H$\alpha$, $L_{H_{\alpha}}/L_{bol}$, and X-ray $L_{X}/L_{bol}$, outputs are seen to decline toward later M spectral types (e.g. Mohanty & Basri, 2003; Berger et al., 2010; Cook et al., 2014), while radio output tends to increase (e.g., Williams et al., 2014), with changes in behaviour appearing to set in around spectral type M7-M8.

Since late M-type dwarfs do not appear to follow the same X-ray behavior as earlier types, we have no indication that their EUV fluxes do either, or how they behave at all. In the absence of an EUV observational capability, other observational and theoretical methods that are able to infer the EUV fluxes of late M dwarfs would be of considerable value. Peacock et al. (2018) have developed theorical atmospheric models that appear promising for constraining emission in the longer wavelengths of the EUV bandpass for very low mass stars. At shorter wavelengths, EUV emission stems from coronal temperatures $\log T\geq 5.5$. While X-ray observations can be used to develop coronal models for temperatures $\log T\geq 6$, there remains a gap in temperature coverage between a few $10^{5}$ K and $10^{6}$ K that is crucial for constraining EUV fluxes.

Here, we develop a method to utilize the Chandra X-ray Observatory High Resolution Camera spectroscopic detector (HRC-S) to provide EUV constraints. This detector has sensitivity well into the EUV range, out to 170 Å. We devise an observing method to utilize an off-axis region of the detector with a thinner Al coating that has higher sensitivity at wavelengths longward of 44 Å than the on-axis filter. We analyse a pilot observation of the late M dwarf LHS 248 and show that the observing method can provide quite tight observational constraints on its EUV flux in the 100-400 Å range. The observing rationale is explained in Sect. 2. In Sect. 3 the various observations utilized in the study are described, and their analysis is related in Sect. 4. We discuss the issues involved in interpreting the data in Sect. 5 and reach our conclusions in Sect. 6.

2 EUV Sensitivity of the Chandra HRC-S

The Chandra X-ray Observatory (Weisskopf et al., 2003) LETG+HRC-S instrument reaches to wavelengths of 170 Å (and even longer with offset pointing), which is well into the EUV part of the spectrum. However, it is essentially a bright object spectrometer: depending on the intervening interstellar medium absorption, spectra are spread over up to 300 mm of microchannel plate detector, and for faint targets the signal becomes background dominated and swamped. With perhaps the exception of our second nearest star, Proxima Cen (M6.5V; see Sect. 4.3.3 below), very late fully convective M dwarfs are generally beyond its reach.

A schematic of the HRC-S UV/Ion shield (UVIS), designed to thwart optical and UV throughput and low-energy protons, is illustrated in Figure 1. The UVIS comprises three segments of Al-coated polyimide film. The two outer segments have the same characteristics within manufacturing tolerances, while the center segment is slightly different. Each filter segment covers a stack of microchannel plates and a charge readout grid connected to signal amplifiers that make up the rest of the detector.

The different segments were designed to handle the different wavelengths of the first order dispersed LETG spectrum. Each has a strip of thicker Al coating, serving as a "low energy" suppression filter and designed for using primarily on the higher dispersed LETG orders throughput, and a larger section of thinner Al as the main first order detection region.

The centre segment, together with the underlying microchannel plate detector, also serves as the back-up imaging detector to the HRC-I, as well as being the prime Chandra instrument for high timing resolution observations. Consequently, the region at the centre where the aim point lies—the vertical part of the “T” shape in Figure 1—has a thicker Al coating for more aggressive UV and optical attenuation. Away from the detector centre, during normal operation with the LETG in place, there is no signal from dispersed UV or optical photons, and the out-of-band rejection requirements are less stringent. Consequently, outside of the “T” region the Al coating is much thinner. This filter region has significantly larger transmittance than the centre region with the thicker Al coating.

The effective areas of the HRC-S “thin” and “thick”’ Al regions at the filter boundary in the $-y$ direction from the aimpoint are illustrated in Figure 2. The differences become pronounced for wavelengths longer than the C K edge near 44 Å, up to the Al L edge near 170 Å, with the area being larger on the thin filter by typical factors of 2–4. It is this difference in effective areas that gives us some power to discriminate between shorter and longer wavelength emission.

3 Observations

Details of the Chandra observations of LHS 248 and Proxima Centuri analysed in this study are listed in Table 1. All data reduction and computation of instrument response files followed standard procedures using the Chandra Interactive Analysis of Observations (CIAO; Fruscione et al. 2006) software framework. Pertinent details for each are described further below.

3.1 LHS 248

LHS 248 (DX Cnc, GJ 1111) has been assigned credible spectral types in the range M6 V (Davison et al., 2015) to M7.1 V (Terrien et al., 2015), with the most common designation being M6.5 V (Alonso-Floriano et al., 2015). At a distance of 3.58 parsecs (Gaia Collaboration, 2018) LHS 248 is the 18th closest star (or star system) to the Sun. It is a rapid rotator and a magnetically active flare star (e.g. Reiners & Basri, 2007), with a period of 0.46 days (Morin et al., 2010; Newton et al., 2018) and a probable member of the Castor moving group with an age of 200 Myr (Caballero, 2010, and references therein). LHS 248 was detected in a 16 ks ROSAT PSPC pointed observation with an X-ray luminosity of $4\times 10^{26}$ erg s*-1* in the survey by Schmitt et al. (1995). The corresponding ratio between X-ray and bolometric luminosities is $\log L_{X}/L_{bol}=-3.9$ (Cook et al., 2014), which is nearly a decade below the canonical saturation threshold of $\log L_{X}/L{bol}\sim-3$ (see, e.g. Wright et al., 2011).

LHS 248 was observed using the Chandra HRC-S on 2018 May 22 for a net exposure of 10 ks. The target was acquired with an offset relative to the on-axis telescope aim point of +5.03 arcmin in order to place the center of the telescope dither pattern on the boundary between the thick and thin Al filter regions.

An important aspect of the observation is that the filter resides above the detector surface by about 12 mm, and a point source at this height above the focal plane has a size of approximately 30 arcsec. The observation therefore employed a special $128\arcsec\times 16\arcsec$ (full width) dither pattern that was enlarged relative to the standard $40\arcsec\times 40\arcsec$ dither in the spacecraft $Y$ axis, along the long axis of the HRC-S, and compressed in the orthogonal spacecraft $Z$ axis111Spacecraft $Y$ and $Z$ axes are aligned with “chip y” and “chip x”, respectively, in HRC-S detector coordinates. This was done so as to minimise the time spent in transit over the thick/thin boundary and to maximise the exposure when the source was entirely either in the thick or thin Al regions.

The basis of the observational method employing the spacecraft dither to sample the stellar emission on the different filters is that frequent sampling better accounts for source variability that can occur on the observation timescale than, for example, dividing the exposure time into two continuous pieces. Ideally, the dither period should be as short as possible. In practice, the shortest dither period achievable is limited by a maximum allowable spacecraft dither speed. In order to accommodate the longer $Y$ axis dither and not violate maximum spacecraft dither rates, the dither period in this axis had to be lengthened from the default 1087s to 2647s. The dither pattern followed during the observation is illustrated in detector coordinates in Figure 3.

The sky image of the detected events is illustrated in Figure 4, together with the circular and annular regions used to extract the source and background signals, respectively. The source photons are spread over a distinctly larger detector area than for sources observed on-axis as a result of the degradation of the telescope point spread function with increasing off-axis angle. The source extraction region has a radius of $5.6\arcsec$, with the annular background region having an area 17.6 times larger.

3.2 Proxima Centuri

Proxima is an M5.5 dwarf with a mass of approximately $0.12M_{\odot}$ (Boyajian et al., 2012; Delfosse et al., 2000, and references therein) and a rotation period of about 83 days (Benedict et al., 1998; Kiraga & Stepien, 2007; Savanov, 2012; Suárez Mascareño et al., 2015). While of slightly earlier spectral type than LHS 248, its proximity has allowed for detailed study from infrared to X-ray wavelengths and, despite its relatively slow rate of rotation, its activity level is not too dissimilar from the latter star. We use it here as a learning example to begin to understand the structure of the coronae of low-mass M dwarfs.

We use high resolution Chandra observations of Proxima Centauri in the analysis described below in order to provide a simple test of our idealized emission measure distribution approach. Proxima was observed using the High Energy (HETG; Canizares et al. 2000) and Low Energy (LETG; Brinkman et al. 2000) transmission grating spectrometers in 2010 and 2017, respectively (see Table 1). The HETG observation used the Advanced CCD Imaging Spectrometer spectroscopic detector (ACIS-S), while the LETG observations employed the HRC-S.

4 Analysis

4.1 LHS 248 Light Curves

The source and background HRC-S light curves of LHS 248 binned at intervals of 200s are illustrated in Figure 5, together with the times at which the source was on thick and thin Al filter regions. This latter calculation assumed that the source was a uniformly illuminated circle and had a radius of $15\arcsec$ at the height of the UVIS above the detector. While the defocused source is likely to be slightly more donut-shaped than a uniformly illuminated circle, this will have only a second order effect on the light curve shape as the source crosses the thick/thin filter boundary, and does not factor into the selection of intervals during which it lies entirely on either the thick or thin Al region.

At approximately 9000s from the start of the observation, a large flare was observed in which the count rate was seen to rise by a factor of about 13 over the quiescent value. This is brought out more clearly in Figure 5 in the light curve when binned at 50s intervals and divided by a factor of 10. Since this flare occurred exclusively on the thin Al filter, we excluded it from the analysis described below.

Figure 5 demonstrates that, while the source was subject to some stochastic variability, the average count rate when on the thin Al filter was always larger than on the thick Al region, as expected.

The source and background light curves are shown as a function of the dither phase in Figure 6. The average source count rates over the whole observation, excluding the flare and time intervals during which the source was in transition between one filter region and the other—i.e. when source photons were simultaneously incident on both sides of the filter boundary—were $0.0294\pm 0.0031$ (thick), $0.0526\pm 0.0040$ (thin), for a count rate ratio thin/thick$\,=1.79\pm 0.24$.

4.2 Assuming an Isothermal Coronal Plasma

The HRC-S detector has essentially no intrinsic energy resolution such that the discrimination between EUV and X-ray photons can only be made by comparing the source count rates in the different filter regions. We proceed by employing optically-thin plasma radiative loss models in order to, firstly, examine how the thick to thin filter count rate ratio is sensitive to the temperature and abundances in an idealized isothermal corona. In Sect. 4.4 below we examine the extent to which different shapes of simple parameterized coronal emission measure distributions informed by available observations are consistent with the data. All the calculations presented here were performed with the IDL222Interactive Data Language ©Harris Geospatial Solutions-based *Package for INTeractive Analysis of Line Emission333PINTofALE is freely available at http://hea-www.harvard.edu/PINTofALE/ * (PINTofALE).

While both solar and stellar coronae are well known to comprise multi-thermal plasmas over a wide range of temperature, it is still useful to examine the extent to which the HRC-S thick and thin Al data are sensitive to isothermal plasma temperature and the assumed chemical abundance mixture. To this end, synthetic spectra were computed within PINTofALE for isothermal temperatures from $10^{5}K$ to $10^{8}K$ using emissivities from the CHIANTI database version 7.1.5. We computed two sets of spectra, one for the solar abundances of Grevesse & Sauval (1998), and the other for an “inverse first ionization potential” (inverse FIP) chemical composition. The latter reflects the now extensive observational evidence that in active stars the abundance of elements with low first ionization potential (FIP; $<10$eV) appear to be depleted relative to elements with high FIP by factors of two or more. This is in contrast to the picture of lower activity stars like the Sun, in which low FIP elements appear to be enhanced (see, for example, Drake et al., 1997; Drake, 2003; Laming, 2015; Wood et al., 2018). These coronal abundance patterns also appear to depend on spectral type, with later K and M stars exhibiting a more inverse FIP pattern and earlier types tending towards a solar-like FIP effect (Wood et al., 2018).

An XMM-Newton study of four active mid-M dwarfs by Robrade & Schmitt (2005) found all to be characterised by an inverse FIP abundance pattern in which low FIP elements such as Mg, Si, and Fe were depleted by about a factor of 2 relative to solar photospheric values while high FIP elements, such as Ne, C, O, and N were relatively enhanced by up to a factor of 2. Robrade & Schmitt (2005) noted that the low FIP deficiency might simply reflect the stellar photospheric metallicity, leaving the high FIP elements enhanced. For our inverse FIP abundance set we adopted a simple pattern in which low FIP element abundances were reduced by a factor of 2 relative to the Grevesse & Sauval (1998) mixture.

Synthetic spectra were convolved with the instrument effective area curves to obtain predicted count rates. We first investigated the effect of interstellar absorption for application to sources in general at arbitrary distances by computing the filter count rate ratios for a range of interstellar medium absorbing columns ranging from $10^{18}$ to $10^{22}$ cm*-2*; ratios for lower column densities are essentially identical to that for $10^{18}$ cm*-2*. The absorbing column toward LHS 248 is not known with any great degree of precision. According to the local interstellar cloud model of Redfield & Linsky (2000)444http://lism.wesleyan.edu/cgi-bin/distlic.cgi, the distance to the edge of the cloud in the line of sight toward LHS 248 is 2.3 pc. They estimate an average neutral hydrogen density of 0.1 atoms cm*-3*, for a column density of $7.1\times 10^{17}$ cm*-2*. Beyond the local cloud in the same sightline lies the Gemini cloud555http://lism.wesleyan.edu/cgi-bin/dynlism.cgi (assuming the structure of the local ISM region comprises discrete clouds; see, e.g., Redfield & Linsky 2015), although there is no strong constraint on its distance. Redfield & Linsky (2008) list a distance of 6.7 pc to the closest star with measured absorption signatures from this cloud. It is therefore possible that there is no further significant absorption toward LHS 248 above that of the local cloud. A range of $5\times 10^{17}$—$3\times 10^{18}$ cm*-2* should then comfortably bracket the actual neutral hydrogen column toward LHS 248.

The predicted count rate ratios for the filters as a function of isothermal plasma temperature for the different ISM absorbing columns, and for the solar and inverse FIP compositions for absorption appropriate to LHS 248 are illustrated in Figure 7.

4.3 Coronal Emission Measure Distributions of M dwarfs

4.3.1 General Considerations

We base the analysis of LHS 248 below on simple parameterizations of the coronal emission measure distribution. In its most simple form, the emission measure as a function of temperature is usually expressed in the form of the differential emission measure (DEM) as a function of temperature, $T$,

[TABLE]

Since the advent of high resolution EUV and X-ray spectroscopy of, first, the solar corona corona and, subsequently, stellar coronae, many studies have examined the form of the plasma DEM. The reader is referred to the following work as a starting point for deeper exploration of the extant literature: Pottasch (1963); Withbroe (1975); Jordan (1976); Craig & Brown (1976); Bruner & McWhirter (1988); Kashyap & Drake (1998); Jordan (2000) for solar work; Drake et al. (1995); Sanz-Forcada et al. (2002, 2003); Huenemoerder et al. (2003); Telleschi et al. (2005); Wood et al. (2018) for example stellar results.

The body of existing work demonstrates some universal aspects of the coronal DEM: from chromospheric temperatures of a few $10^{4}$ K the DEM decreases by approximately an order of magnitude to a minimum at temperatures of approximately 1–$4\times 10^{5}$ K; the DEM then rises by a approximately order of magnitude, or more, to a maximum at temperatures between $10^{6}$–$10^{7}$ K, beyond which it extends to higher temperatures over either a plateau or a shallow downward slope, before a precipitate decline by several orders of magnitude. It is possible there is further fine structure in the shape of the DEM in some cases, although assessing the veracity of such structure is far from trivial owing to the nature of the ill-constrained integral inversion problem of inferring the DEM from observed spectra (see, e.g., Craig & Brown, 1976; Kashyap & Drake, 1998).

4.3.2 The case of Proxima: Chandra HETG Spectrum

While existing stellar DEM data exhibit common characteristics, there is essentially no detailed spectral information for late M dwarfs with which to verify that such characteristics do extend to the lowest mass stars. Here, we examine the Chandra HETG spectrum of Proxima Centuri in order to help establish the form of emission measure distribution to adopt for our target LHS 248. The analysis employed a parameter estimation approach in which the observed spectrum was fitted to a multi-thermal, optically-thin plasma radiative loss model whose emission measure distribution at each temperature was allowed to vary.

Proxima is the most well-known flare star in the sky, and in order to understand the context of the extracted spectra upon which our analysis is based we first extracted the dispersed photon events in 0th order to compute the X-ray light curve and assess flaring activity. An approximate correction for the effects of “pile-up”—when photon events are lost or incorrectly counted due to multiple photon interactions at the same detector location during a single CCD frame—using pile-up fraction estimates from the Portable, Interactive Multi-Mission Simulator666http://cxc.harvard.edu/toolkit/pimms.jsp was made to the extracted light curve. The resulting corrected light curve is illustrated in Figure 8. Pile-up fractions varied from a maximum of approximately 30% for the very peak of the flare at about 20 ks into the exposure to a minimum of about 2% for the lowest count rates and a more typical 6% for count rates of $\sim 0.05$ count s*-1*. The observation is characterised by a moderate, and likely typical, amount of flaring (see, eg, the survey of Proxima X-ray observations by Wargelin et al., 2017), in which several events are observed to reach count rates an order of magnitude above the apparent quiescent rate.

Parameter estimation was undertaken using the Sherpa fitting engine within CIAO. Multiple Astrophysical Plasma Emission Code (APEC777http://www.atomdb.org/) thermal plasma models were adopted to represent the DEM on a fixed temperature grid, each with a common fixed abundance pattern whose absolute normalization—the metallicity—was free to vary. The temperature grid spanned the range $\log T=6.25$–7.5 in intervals of 0.25, for a total of 6 thermal plasma components.

While the abundances could in principle be fitted simultaneously with the temperature structure, we found that the significantly larger number of free parameters required posed problems for the stability of the solution. The solar abundances of Grevesse & Sauval (1998), modified to approximate the “inverse FIP” pattern by increasing by a factor of two the abundances of C, N, O, Ne and Ar, as described above in Sect. 4.2, were adopted.

Since the high-resolution Chandra spectra of Proxima comprise many bins with few counts, the Cash statistic ($\mathcal{C}_{r}$; Cash 1979) was employed for minimization of model deviations from the data. This statistic is valid in the Poisson regime of low numbers of counts in which $\chi^{2}$ approaches, relying on Gaussian uncertainties, are inapplicable. This allows the data to be analyzed without further grouping of neighboring bins, and at full spectral resolution.

The HETG High Energy Grating (HEG) spectrum was fitted over the wavelength range 1.5–16 Å, while fits to the Medium Energy Grating (MEG) spectrum were restricted to 2–25 Å. Both HEG and MEG spectra were initially fit simultaneously, but the low signal-to-noise ratio of the HEG spectrum lead to difficulties in obtaining convergence and final adopted results were based on the best-fit to the MEG spectrum. The best-fit model yielded a reduced statistic of 0.6 and a formal metallicity of $[M/H]=-0.37\pm 0.04$, expressed in the conventional logarithmic bracket notation.

The best-fit model is illustrated superimposed on the MEG spectrum in top panel of Figure 9, while the resulting emission measure distribution is shown in Figure 10.

4.3.3 Proxima Cen LETGS spectrum

A global model fit of the Chandra LETGS spectrum similar to that performed for the HETG unfortunately proved infeasible due to the low signal-to-noise ratio in the vast majority of the spectral bins. However, it was instead immediately instructive to simply superimpose the spectrum computed for the HETG best-fit parameters. This overlay is illustrated in Figure 9. Note that no scaling or adjustment to any parameters has been made, except to scale back the C and N abundances to their solar ratios, [C/Fe]=0 and [N/Fe]=0 for illustrative reasons explained below. The agreement at wavelengths $<25$ Å is quite remarkable, indicating that the average activity level of the two observations, flares included, was essentially the same.

The coolest temperature constrained by the HETG data was an upper limit to the DEM at logarithmic temperature $\log T=6.25$. The longer wavelengths of the LETGS spectral range exhibit lines formed at cooler plasma temperatures than sampled in the HETG range and we can in principle use these lines to better constrain the lower temperature DEM. A glance at the centre panel of Figure 9 reveals a slew of emission lines longward of 50 Å in the model spectrum that appear in generally good agreement with the observed spectrum in the sense that they at least do not exceed the observed flux.

We next illustrate two similar spectral models computed with cool extensions to the HETG DEM that are illustrated in Figure 10. The form of the DEM extension is fairly arbitrary but based on the generally observed trends noted in Sect. 4.3.1; we simply extrapolate from the trend in lower-T behavior seen here and found in M dwarfs by Robrade & Schmitt (2005, see Sect. 4.4 and Figure 10), using $DEM\propto T^{\alpha}$, with $\alpha=3/2$ and 5/2. The spectrum corresponding to a shallower power law slope, $\alpha=3/2$, egregiously over-predicts several lines, most notably the O VI doublets at 104.8 Å and 150 Å, and Ni X and XI at 148.6 and 148.4 Å, respectively. Instead, the spectrum corresponding to the steeper DEM slope, $\alpha=5/2$, is in quite good agreement with the lines in the longer wavelengths of the LETG spectrum.

Returning to the elements C and N, whose abundances were not scaled to the inverse FIP pattern adopted for the HETG spectrum fitting for the spectra illustrated in Figure 9, we note that both their H-like doublets at 33.7 Å and 24.8 Å denoted in the lower three panels are under-predicted by a factor of 2 in the $\alpha=5.2$ spectrum. The He-like resonance lines at 28.8 and 40.3 Å are similarly under-predicted. This remains the case even for the elevated cool DEM $\alpha=3.2$ spectrum, although to a slightly lesser extent. These lines are formed at 1–$2\times 10^{6}$ K and their under-prediction indicates that the inverse FIP abundance pattern extends to cooler temperatures than represented by the bulk of the HETG spectrum.

4.4 A grid of model DEMs for LHS 248

The general emission measure distribution shape that matches Chandra HETG and LETG spectra of Proxima reasonably well comprises a fairly steep power law slope from $\log T=5.5$ until a peak at $\log T=6.5$, a shallow power law-like decline to $\log T=7.25$, followed by a steep drop off toward higher temperatures. By way of an interesting comparison to Proxima, Figure 10 also illustrates the average emission measure distribution for the four active mid-M dwarfs (AD Leo, EV Lac, AT Mic and EQ Peg) whose XMM-Newton spectra were analysed by Robrade & Schmitt (2005). The average of these M dwarf DEMs can be well-approximated, at least within the temperature range probed by the XMM-Newton data, by two power law components: a cool component rising toward a the temperature of peak emission measure and a declining power law toward higher temperatures.

Inspired by these similar-looking general DEM characteristics, and lacking detailed spectral information, we proceed to analyse the LHS 248 HRC-S photometry using a grid of model DEMs with a shape guided by the observed DEM characteristics. The general form adopted is

[TABLE]

where $T_{min}$ is the temperature at which the DEM reaches a minimum, $T_{peak}$ is the temperature at which the DEM peaks, $T_{sh}$ is the maximum temperature reached by the DEM “plateau” at its shoulder region, and $\alpha_{T}$, $\alpha_{LC}$, $\alpha_{UC}$ and $\alpha_{HT}$ are the power law indices for the transition region, the “lower corona”, the “upper corona” and the high temperature tail, describing the slopes to the DEM in the different temperature regions. The value of $\alpha_{LC}$ is always positive, $\alpha_{T}$ and $\alpha_{HT}$ are always negative, and $\alpha_{UC}$ could in principle be either positive or negative but in our model DEMs takes on a fixed negative value.

In order to limit the number of parameters in the analysis to a manageable set, we use the observed DEMs in Figure 10 and appearing in the literature (Sect. 4.3.1) to constrain the general shape and parameter range to investigate. The temperature of the DEM minimum was assumed to be $\log T_{min}=5.5$, below which a fixed power law was assumed for all DEM models. Plasma at temperatures below $\log T=5.5$ does not contribute significant emission in our HRC-S bandpasses in the presence of hotter coronal emission and the details of this cooler part of the DEM are not important for our analysis. The full set of fixed constraints are

[TABLE]

The parameters that were allowed to vary are the temperature of the peak DEM, $T_{peak}$, and the amplitude of the DEM peak relative to the DEM at $T_{min}$, specified by the scaling factor relating $\Phi(T_{peak})$ and $\log\Phi(T_{min})$. These relations are as follows:

[TABLE]

The set of model DEMs investigated are illustrated in Figure 11. For each DEM, the synthetic spectrum was computed within the PINTofALE framework for the adopted inverse FIP abundance pattern and the ratio of thick to thin filter count rates was calculated. DEMs for which the count rate ratio was in agreement with the $1\sigma$ uncertainty of the measured value are highlighted.

5 Discussion

The off-axis pointing mode aimed at the HRC-S thick/thick Al filter boundary yielded photometric results for LHS 248 in agreement with general expectations: the count rate through the thin Al filter was found to be significantly larger than through the thick Al filter region owing to the larger effective area in the $\sim 40$–170 Å range in the former. The analysis applied above suggests that this extra soft X-ray throughput is perhaps larger than might have been expected, since the observed ratio is $\sim 1.8$ while ratios from our model DEM predictions top out below 1.7.

5.1 Isothermal Analysis

The isothermal analysis illustrated in Figure 7 suggests the data are compatible with a rather cool corona with a temperature $\log T<6.4$, quite similar to the dominant temperature of the solar corona. Such a coronal temperature would be at odds with the rapid rotation of LHS 248 and its quite high X-ray luminosity in relation to its bolometric output, $L_{X}/L_{bol}=-3.9$. In the presence of multi-thermal plasma, this simplistic interpretation of Figure 7 is of course not valid. An admixture of hotter and cooler plasma could also reproduce the observed count rate ratio.

The peak in the synthetic ratio at $\log T=5.6$–5.7 indicates that plasma at that temperature has a particularly strong influence on the relative count rates in the different filters. This is largely due to the fall off in the short wavelength end of the spectrum such that the dominant contribution to both filter count rates shifts longward of the C K edge. The flatter ratio for temperatures $\log T>6.5$ is due to the opposite effect and increasing dominance of the spectrum at shorter wavelengths where the filter transmittances are similar. The influence of the abundance pattern—inverse FIP versus a solar photospheric mixture—makes little difference to the count rate ratio.

The small values of interstellar hydrogen column density toward the most nearby stars has similarly weak influence. In general, the sensitivity of the filter ratio is expected to gradually diminish with increasing interstellar medium absorption as the longer wavelengths of the spectrum are increasingly attenuated. However, the picture is complicated by strong lines of Fe IX-XI in the 170–200 Å range and longward of the Al L edge where the two filter transmittances are very similar. Hence the curves in the upper panel of Figure 7 show quite a complicated pattern of predicted count rate ratio until column densities reach about $10^{20}$ cm*-2* at which point further increases in column yield the expected behaviour.

5.2 How steep is the DEM?

The surprising result from the model differential emission measure distribution analysis is that shallow DEM slopes are favored by the LHS 248 data, in constrast to the results from the Proxima spectra. The thin/thick count rate ratio is illustrated for the model DEMs as a function of their power law slopes from the emission measure minimum to the peak, $\alpha_{LC}$ in our model notation, in Figure 12. It is clear that no model is strongly excluded, since all satisfy the observations within the $2\sigma$ limit (the observation analysed here represents a pilot study; a longer exposure would have helped in providing more stringent constraints). Figure 12 indicates that slopes of $\alpha_{LC}\leq 3/2$ are favoured.

The value of the slope of the coronal emission measure has been examined in detail dating back to earlier solar studies. Those studies found values close to $\alpha_{LC}=3/2$ (e.g. Athay, 1966; Jordan, 1975, 1976), a result that can be understood in terms of the energy balance in spherically-symmetric hydrostatic equilibrium and constant cross-section loop models considering the dissipation of heating within the corona and cooling through radiation and conduction back to the chromosphere. Extensive discussions have been provided by, e.g., Jordan (1976); Craig et al. (1978); Rosner.etal:78; Jordan (1980); van den Oord et al. (1997) and more recently by Jordan et al. (2012).

From a purely empirical perspective, the coronal emission measure distribution based on EUV lines observed in EUVE spectra is seen to steepen in more active stars, with values of $\alpha_{LC}\sim 2$–4 having been derived (e.g. Laming et al., 1996; Laming & Drake, 1999; Sanz-Forcada et al., 2003), similar to the cores of the brightest active regions on the Sun (Drake et al., 2000). The steeper slopes are possibly the result of flare rather than steady heating (e.g. Güdel, 1997; Audard et al., 2000; Kashyap et al., 2002). Similar steeper slopes for active single and binary stars of a variety of spectral types were subsequently derived from X-ray observations with Chandra and XMM-Newton diffraction gratings (e.g., beginning with Drake et al., 2001; Huenemoerder et al., 2001, 2003, 2006; Telleschi et al., 2005). We therefore expect the DEM to be somewhat more steep than $\alpha_{LC}\sim 3/2$. This expectation is of course predicated on extrapolation of the behaviour of higher mass stars to late M dwarfs.

The different chromospheric and coronal behaviour of very low mass stars compared with stars of mid-M spectral type and earlier was briefly referred to in Sect. 1. The fractional X-ray output, $L_{X}/L_{bol}$, of the lowest mass stars is systematically lower than for higher masses (e.g. Fleming et al., 2003), exhibits a strong enhancement of the “supersaturation” effect—that X-ray output decreases with increasing rotation rate for the fastest rotators—and the scatter in X-ray output at a given rotation period is three times larger than for higher mass stars Cook et al. (2014). Chromospheric Hα fluxes are also relatively depressed (Delfosse et al., 1998; Mohanty & Basri, 2003; Fleming et al., 2003). In contrast, the radio output of the lowest mass stars is higher than expected based on the behaviour of stars of higher mass (Berger, 2006; Berger et al., 2008; Williams et al., 2014).

The generally favoured explanation for these changes in behaviour is a speculated change in the underlying magnetic dynamo in fully-convective stars and the form of surface magnetic field that is produced, perhaps coupled with the effects of a very low electrical conductivity in the quasi-neutral atmospheres. Wright & Drake (2016) have shown that there is no obvious change in X-ray behaviour across the fully-convective limit, with slowly rotating fully-convective stars like Proxima and Barnard’s Star adhering reasonably closely to the rotation-activity relation of stars with a central radiative zone. The results were further bolstered with a larger sample by Wright et al. (2018). The latest spectral type of the slow rotators in the Wright et al. (2018) sample is M6—comparable to the type at which the magnetic activity indicators appear to change in behaviour. Thus the existing X-ray-rotation studies do not provide any further strong clues as to the nature of the dynamo in the latest M types.

In other aspects, the characteristics of the X-ray emission of late M dwarfs is not conspicuously different to their earlier counterparts, though there are indications of a possible decline in coronal temperature. While Wheatley et al. (2017) found the X-ray emission from the M8 dwarf TRAPPIST-1 could be approximated with a two-temperature optically-thin plasma radiative loss model with temperatures of 0.15 and 0.83 keV, Fleming et al. (2003) found the temperature of the corona of the M8 dwarf VB 10 to be only 0.24 keV.

5.3 Missing lines in the 30–170 Å range?

Any interpretation of X-ray and EUV photometry in terms of plasma radiative loss models relies to some extent on the completeness of the model spectrum. There are indications that plasma models are still incomplete in the 30-170 Å range. This “missing lines” problem was first raised in the context of the analysis of EUVE spectra: global model fits to spectra in the 70-170 Å range found weaker line-to-continuum ratios than expected that were interpreted as either optical depth due to resonance scattering, or else low metal abundances, or very hot plasma that contributed only continuum to the region. The explanation is instead that flux in lines absent from the models was misinterpreted as continuum (see, e.g., the discussions in Drake, 1996; Schmitt et al., 1996; Drake et al., 1997).

Before the launch of Chandra, the 30–70 Å spectral range covered by the LETGS had also been seldom observed and models were relatively poorly tested in this region. Testa et al. (2012) used the Chandra LETGS spectrum of Procyon to evaluate atomic data relevant to the narrow-band filters of the Solar Dynamics Observatory and concluded that the CHIANTI model (Version 6) was missing flux amounting to up to a factor of 3 relative to the observed spectrum in some regions of the 50–130 Å range. Testa et al. (2012) noted other databases shared the same problem. The culprit appears to be some missing transitions from ions of abundant elements such as Ne, Mg, Si, S and Ar with $n=2$ ground states, and from Fe ions with $n=3$ ground states (e.g. Jordan, 1996; Lepson et al., 2005; Liang & Zhao, 2010; Testa et al., 2012).

These missing lines are more prominent in cooler coronal spectra—the DEM for Procyon peaks around $\log T\sim 6.3$, for example (e.g. Drake et al., 1995). Since the missing model flux is greater in the spectral region where the Thin filter has a significantly larger effective area than that Thick region, the effect would be to underestimate the Thin/Thick ratio for a given emission measure distribution. In order to compensate for this, the DEM would need to be more shallow so as to have more cooler plasma that can mimic the missing flux. It is possible, then, that the relatively shallow DEM slopes favoured by the LHS 248 observation are at least partly an artifact of deficiencies in the radiative loss model.

5.4 Utility of the Thin/Thick ratio for predicting the EUV flux

The ultimate goal of the experiment was to design an observation that would provide information to help assess coronal emission in the 100-900 Å EUV spectral range of stars that are too faint to be observed with the Chandra LETGS. To provide an illustration of the EUV leverage of our observations of LHS 248, the set of X-ray–EUV spectra predicted by the DEM models with a temperature at DEM maximum of $\log T_{peak}=6.5$ and binned on 20 Å intervals (for other values of $T_{peak}$ within the same confidence range spectra are very similar) are illustrated in Figure 13.

The EUV flux shortward of about 400 Å is dominated by temperatures $\log T>5.5$, a regime for which our modelling approach should be able to provide an accurate description. Toward longer wavelengths, emission is dominated by cooler plasma and our assumptions regarding the emission measure at its minimum, $\Phi(T_{min})$, and at cooler temperatures, $\Phi(10^{4}\,K)$, begin to be important. Additionally, large contributors to the flux in the 800–900 Å and 450-500 Å ranges are the H Ly$\alpha$ and He I continua that are formed largely in the chromosphere and lower transition region (see also, e.g., Linsky et al., 2014). While our DEM models extend down to chromospheric temperatures, assumptions of optically-thin, collision-dominated thermal equilibrium tend to break down there and reproduction of the recombination continuum is not likely to be wholly accurate.

Proceeding with the assumption that missing lines do not have a significant impact on the photometric filter ratio interpretation, Figure 13 demonstrates that the general uncertainty in the flux at 100 Å within the $1\sigma$ range given by the spread in the model fluxes is only about 25%. Uncertainties grow, as would be expected, toward longer wavelengths. In the 200–300 Å region the $1\sigma$ range is a factor of 2–3. The flux at 300 Å is dominated by the He II Ly$\alpha$ transition at 303.78 Å whose formation in the chromosphere, as for the H I and He I continua, is likely subject to complications from photoionization and recombination and thus might not be accurately represented by optically-thin collision-dominated models in thermal equilibrium (e.g. Jordan et al., 1993, and references therein). The flux uncertainty grows to a factor of 4–5 in the 300-400 Å range, and beyond that is about an order of magnitude.

These considerations lead us to conclude that the observational method developed in this paper can provide a reasonably accurate extrapolation of the EUV flux of stars. Given observations of higher S/N than attained for LHS 248, tight constraints on EUV emission in the 100-400 Å range are achievable. The efficacy of the filter ratio method is somewhat dependent on the completeness of radiative loss models; a re-examination of this issue would be strongly motivated.

Figure 13 also illustrates the EUV-Ly$\alpha$ scaling relations derived by (Linsky et al., 2014) that are in common use. There appear to be no existing Ly$\alpha$ measurements for LHS 248, and so we have simply scaled the flux for Proxima by the relative surface areas, $R_{248}/R_{Prox}$, to set the normalization. The agreement with our models shortward of 400 Å is good for the models with steeper DEM slopes. However, the scaling relations predict a strong discontinuity in the EUV flux at 400 Å by almost a factor of 100, in the sense that the flux in the 400-600 Å range appears to be strongly underpredicted. It can be seen from our spectral models that such a discontinuity is unphysical in the sense that no superposition of plasma temperatures can conspire to give a spectrum with such a large jump in the flux at that wavelength. The minimum flux in the 400-600 Å range is given by purely hot coronal emission with temperatures in excess of $10^{6}$ K that are adequately constrained by X-ray observations. Plasma at temperatures $<10^{6}$ K only adds to this.

6 Conclusions

A specially-designed off-axis Chandra HRC-S observation of the late M dwarf LHS 248 succeeded in placing the source on the boundary between Thin and Thick Al coating regions of the UV/ion shield filter. Dithering the source between the two regions of the filter resulted in a modulation of the observed count rate such that the rate for the Thin filter was consistently higher than for the Thick. The count rate difference is due to the effective area of the Thin filter being larger by up to a factor of 4 over that of the Thick filter in the 44–170 Å range.

A set of model DEMs based on the DEM shape found from an analysis of an archival Chandra HETG spectrum of Proxima and also guided by results in the literature was generated. These DEMs were used to compute model spectral that were assessed according to their ability to reproduce the observed Thin/Thick count rate ratio. Shallow power law slopes to the DEM in the temperature range $5.5\leq\log T\leq 6.5$ are favoured at the $1\sigma$ level, but steeper slopes are accommodated within a $2\sigma$ range. It is possible that the shallow slopes are an artifact of incompleteness in current radiative loss models in the 30–170 Å range; re-examination of this issue would be strongly motivated.

The modelling approach adopted here indicates that, in principle, Chandra HRC-S Thin/Thick photometric observations can provide an accurate—at least within a factor of 2-4—estimate of the EUV fluxes of stars in the 100-400 Å range. Constraints at longer wavelengths require more accurate knowledge of the cooler $\log T<5.5$ emission measure. We also find that the commonly used EUV scaling relations for M dwarf spectral types of Linsky et al. (2014) have an unphysical discontinuity at 400 Å and likely underestimate the flux in the 400-600 Å range.

This work was funded by NASA Chandra grant GO8-19015X and NASA contract NAS8-03060 to the Chandra X-ray Center. JJD thanks the Director, Belinda Wilkes, for continuing advice and support. Chandra X-ray Observatory